Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity

In this paper we consider the Cauchy problem for a generalized Camassa-Holm equation with cubic nonlinearity in Besov spaces. We first establish the local well-posedness of the equation in the Besov space Bp,rs by using the Littlewood-Paley theory. Then, under a sign condition we reach the sign-preserved property and a precise blow-up criterion. Applying this precise criterion we present a blow-up result and the precise blow-up rate for strong solutions to the equation.